1 00:00:01,490 --> 00:00:06,917 Until now we have been studying mostly quantum theories and physical phenomena 2 00:00:06,917 --> 00:00:10,348 which were discovered relatively a long time ago. 3 00:00:10,348 --> 00:00:15,613 So I think maybe the latest theoretical development we talked about, the path 4 00:00:15,613 --> 00:00:18,860 integral itself goes back to the 40's, 1940's. 5 00:00:18,044 --> 00:00:23,840 Today in this video, towards the end of the video I'm going to tell you a, about a 6 00:00:23,840 --> 00:00:29,380 more recent theoretical development. A so-called theory of quantum localization 7 00:00:29,380 --> 00:00:35,626 or more precisely weak localization. And that will do so using the main ideas 8 00:00:35,626 --> 00:00:40,184 of the Feyman path integral. So I should mention that the 4 theory of 9 00:00:40,184 --> 00:00:44,807 quantum localization is an extremely complicated mathematical theory and it 10 00:00:44,807 --> 00:00:47,480 could be a subject though for separate course. 11 00:00:47,480 --> 00:00:51,014 So I don't have time for that and obviously, we don't have the mathematical 12 00:00:51,014 --> 00:00:55,837 sort of background to talk about it. But it turns out that some basic ideas 13 00:00:55,837 --> 00:01:01,492 behind this phenomenon can be understood using the basic concepts behind the 14 00:01:01,492 --> 00:01:04,981 Feynman path integral, which comes handy here. 15 00:01:04,981 --> 00:01:10,403 So just in a few words, what I'm going to be taking about here is the following. 16 00:01:10,403 --> 00:01:15,695 So we know that in solid state physics, materials sort of can be divided into 17 00:01:15,695 --> 00:01:19,791 roughly two categories. Insulators and metals. 18 00:01:19,791 --> 00:01:24,740 They're also semiconductors but from this perspective, they're much like insulators. 19 00:01:24,740 --> 00:01:29,162 So and if you, what insulators are, if you apply a weak electric field the electrons 20 00:01:29,162 --> 00:01:33,127 which exist inside the material don't move and so there is no current really in 21 00:01:33,127 --> 00:01:36,992 response to electric field. So, If you think about a piece of food or 22 00:01:36,992 --> 00:01:40,186 something like that. So we obviously understand it's an 23 00:01:40,186 --> 00:01:43,347 insulator, it's not going to be a very useful electric component. 24 00:01:43,348 --> 00:01:47,707 On the other hand, if you have a metallic system, let's say a piece of copper, and 25 00:01:47,707 --> 00:01:51,991 if we apply electric field or voltage there will be current in response to this 26 00:01:51,991 --> 00:01:57,311 electric field. And so this characterizes a metal, so in 27 00:01:57,311 --> 00:02:05,150 [unknown] there is some intersections disorder, impurities , dislocation all 28 00:02:05,150 --> 00:02:10,970 kind of things and these imperfections. They lead to a finite resistance. 29 00:02:10,970 --> 00:02:11,611 Okay, and so, the next slide will tell you about how this resistance comes about. 30 00:02:11,611 --> 00:02:18,145 But later on we will see how essentially the classical theory of transport that 31 00:02:18,145 --> 00:02:24,679 applies to most metal at high temperature get more defined as you go to lower and 32 00:02:24,679 --> 00:02:31,708 low temperatures quantum effects come into play, the way of nature for instance, the 33 00:02:31,708 --> 00:02:37,945 play, and this is very interesting and it turns out that there under certain 34 00:02:37,945 --> 00:02:44,083 circumstances you can get complete completely different behavior at low 35 00:02:44,083 --> 00:02:48,153 temperatures. Metal actually would become insulators and 36 00:02:48,153 --> 00:02:52,642 this process of trapping electrons due to quantum mechanical effects is called 37 00:02:52,642 --> 00:02:56,160 mobile localization. Of course, it's a very mathematical 38 00:02:56,160 --> 00:03:00,820 theory, as I told you, but we will try to get some idea about where it comes from. 39 00:03:00,820 --> 00:03:06,085 But before discussing a complicated quantum theory, let me first talk about a 40 00:03:06,085 --> 00:03:11,836 basic classical, very simple, theory of how we can understand electrical transport 41 00:03:11,836 --> 00:03:16,799 in a metal. So here this black dot labels an electron 42 00:03:16,799 --> 00:03:19,937 which can move around in, in a piece of metal. 43 00:03:19,938 --> 00:03:24,990 And these red dots here, large dots, they represent some impurities, both of which 44 00:03:24,990 --> 00:03:29,411 the electron can scatter. And that's exactly what it does, as it 45 00:03:29,411 --> 00:03:34,452 moves around so it gets scattered by these random impurities and soon the trajectory 46 00:03:34,452 --> 00:03:38,591 of this electron is this diffused trajectory, so it's random walk. 47 00:03:38,592 --> 00:03:42,098 Of course it's not just one, there are many, many electrons in the metal and they 48 00:03:42,098 --> 00:03:46,013 all experience this random walk. Which in the absence of an electric field 49 00:03:46,013 --> 00:03:48,973 just averages out to zero. There is no net current, there is no 50 00:03:48,973 --> 00:03:51,760 preferential direction, there is nothing which breaks the symmetry. 51 00:03:51,760 --> 00:03:55,890 But now, if we let's say apply electric field in this direction the electrons 52 00:03:55,890 --> 00:03:59,489 which have a negative charge are going to move in the opposite direction 53 00:03:59,489 --> 00:04:02,782 preferentially. They are still going to move around in all 54 00:04:02,782 --> 00:04:07,397 directions, but there will be sort of a drift in one particular direction, and 55 00:04:07,397 --> 00:04:12,586 this will give rise to an a current. Now it turns out that the first theory of 56 00:04:12,586 --> 00:04:17,130 this conductivity of this current in a metal was put together back in 1900 and 57 00:04:17,130 --> 00:04:21,745 this [unknown] theory is the so called Drude theory which is a very simplistic 58 00:04:21,745 --> 00:04:26,885 theory that I am going to explain now. So, basically the Drude model assume that 59 00:04:26,885 --> 00:04:32,099 we can just view the [unknown] as a classical particle which there [inaudible] 60 00:04:32,099 --> 00:04:36,485 was the classical equations of motion, the Newton equation. 61 00:04:36,486 --> 00:04:41,040 And in the presence of electric field so that the charge, let's see if the charge 62 00:04:41,040 --> 00:04:45,726 is q, so it experiences this force due to the electric field, and these impurities 63 00:04:45,726 --> 00:04:49,867 all these imperfections, he just replaced with the friction force. 64 00:04:49,868 --> 00:04:57,842 And this friction force was assumed to be proportional to the velocity. 65 00:04:57,843 --> 00:04:58,111 Lets say, it sort of makes sense if you have a viscous fluid for instance, and if 66 00:04:58,111 --> 00:05:00,170 you put. An object is going to move in this fluid 67 00:05:00,170 --> 00:05:04,070 so its going to experience some force, a friction which is proportional eh 68 00:05:04,070 --> 00:05:08,360 proportional to the velocity and with the minus sign of course and there is some 69 00:05:08,360 --> 00:05:11,322 coefficient. So velocity is momentum divided by the 70 00:05:11,322 --> 00:05:15,741 mass. So we can write instead of writing the 71 00:05:15,741 --> 00:05:24,518 friction force this way we can write it as minus the p over t of some tau which is 72 00:05:24,518 --> 00:05:31,198 the time scale which describes. When we have the [unknown] in this prose. 73 00:05:31,198 --> 00:05:36,448 So in some sense you may think about the stall as a typical time between collision 74 00:05:36,448 --> 00:05:38,948 for the electron. But in the drude model it was just a sort 75 00:05:38,948 --> 00:05:41,654 of in the logical parameter of the described friction, or momentum relaxation 76 00:05:41,654 --> 00:05:45,152 of this electron. And now what he did, he simply wrote down 77 00:05:45,152 --> 00:05:52,124 this model, and so of course we can write the left-hand side of the Newton equation 78 00:05:52,124 --> 00:05:57,836 as dp/dt, and so the right-hand side just putting this together is going to be the 79 00:05:57,836 --> 00:06:03,164 electric force and this friction force. And in equilibrium, we demand that there 80 00:06:03,164 --> 00:06:07,030 is no acceleration of electrons. So if we, sort of, if we, if we take let's 81 00:06:07,030 --> 00:06:10,726 say, a piece of metal, apply some electric field, eventually it's going to come to 82 00:06:10,726 --> 00:06:17,588 some sort of state of equilibrium. There's going to be a current, but there's 83 00:06:17,588 --> 00:06:24,495 not going to be indefinite acceleration of electrons. 84 00:06:24,495 --> 00:06:25,140 So they're going to come to a certain sort of steady state, and this steady state 85 00:06:25,140 --> 00:06:28,948 will determine the current. Now and from here what we can do we can 86 00:06:28,948 --> 00:06:35,838 just solve for, for the velocity so momentum is equal from this equation is 87 00:06:35,838 --> 00:06:42,622 going to be equal q times E times tal and velocity therefore is going to be just 88 00:06:42,622 --> 00:06:47,352 momentum divided by the map. And at this stage what we can do, so we 89 00:06:47,352 --> 00:06:51,309 can write the current. So, the current is simply the charge of, 90 00:06:51,309 --> 00:06:56,919 of the carriers in this case electrons, times the density of these carriers how 91 00:06:56,919 --> 00:07:02,529 many of those participate in transport and v is the velocity, which we have just 92 00:07:02,529 --> 00:07:05,088 determined. And so, if we put everything together, 93 00:07:05,088 --> 00:07:07,100 we're going to get this expression. So, it's going to be this [unknown] of 94 00:07:07,100 --> 00:07:08,248 friction, which is called conductivity and times the electric field. 95 00:07:08,248 --> 00:07:16,512 So, by definition actually the conductivity is the coefficient of 96 00:07:16,512 --> 00:07:23,808 proportionality between the current and the electric field. 97 00:07:23,808 --> 00:07:24,150 So this expression, this nq squared tau over m is the so-called drude 98 00:07:24,150 --> 00:07:24,602 conductivity, which as you can see is a purely classical entities. 99 00:07:24,602 --> 00:07:27,421 So there is no quantum mechanics here whatsoever. 100 00:07:27,421 --> 00:07:37,273 Now, embarrassing thing that I have to admit at this stage, is that almost all 101 00:07:37,273 --> 00:07:46,681 sort of useful theories of metals, of transporting metals rely on this drude 102 00:07:46,681 --> 00:07:49,834 equation up to this date. So since, for more than 100 years we've 103 00:07:49,834 --> 00:07:49,895 been using this equation and in most cases actually it works fine. 104 00:07:49,895 --> 00:07:49,958 But fortunately well for us scientists otherwise it would have been really 105 00:07:49,958 --> 00:07:52,332 boring. So if you go to lower temperatures, the 106 00:07:52,332 --> 00:08:04,192 oxidation becomes a little more tricky. And this is because quantum mechanics 107 00:08:04,192 --> 00:08:10,978 become, becomes important. So quantum mechanics come into play. 108 00:08:10,978 --> 00:08:11,040 And what it means is that, in this picture, for instance, what, what, how we 109 00:08:11,040 --> 00:08:11,107 can envision the appearance of quantum effects is that instead of just straight 110 00:08:11,107 --> 00:08:11,204 trajectories of classical particles, we're going to have waves. 111 00:08:11,204 --> 00:08:16,058 Bouncing ground of, of this incurities and so this waves are going to interfere with 112 00:08:16,058 --> 00:08:25,010 one at, with each other in some sense. So its going to be the particles going to 113 00:08:25,010 --> 00:08:38,985 interfere with itself and this quantum interference [inaudible] are going to, 114 00:08:38,985 --> 00:08:47,891 going to modify this picture a very significantly. 115 00:08:47,891 --> 00:08:52,679 So now let us look at the mo, electron motion in methalanic disorder, a metal 116 00:08:52,679 --> 00:08:57,695 from a slightly different perspective, actually from a completely different 117 00:08:57,695 --> 00:09:03,040 perspective because now we want to exclusively to chernon quantum mechanics. 118 00:09:03,040 --> 00:09:07,600 And we will use quantum mechanics [unknown] of quantum mechanics due to 119 00:09:07,600 --> 00:09:12,616 Feynman and ask the question of what is the probability of for an electron to 120 00:09:12,616 --> 00:09:17,890 diffuse from an initial point into the final point F for this forest of impurity? 121 00:09:17,890 --> 00:09:23,183 And according to the Feynman this probability can be written as the absolute 122 00:09:23,183 --> 00:09:28,476 value squared of a sum of quantum mechanical amplitudes associated with all 123 00:09:28,476 --> 00:09:32,851 possible classical paths. So this diffusion trajectory here, I'm 124 00:09:32,851 --> 00:09:35,646 just showing two such possible trajectories. 125 00:09:35,646 --> 00:09:41,046 You know this blue one and the black one and this guys are going to interfere with 126 00:09:41,046 --> 00:09:46,646 one another and this interference suppose instead if I call this trajectory, 127 00:09:46,646 --> 00:09:51,926 trajectory 1, and this is going to be trajectory 2 so, the interference term 128 00:09:51,926 --> 00:09:57,446 which is which is going to appear due to this from this expression is going to 129 00:09:57,446 --> 00:10:02,326 involve a product of this black exponential [unknown] h bar times the Lew 130 00:10:02,326 --> 00:10:07,766 exponential which is complex conjugate [unknown] divided h bar plus its complex 131 00:10:07,766 --> 00:10:11,362 conjugate. So, oh no, the interference term coming 132 00:10:11,362 --> 00:10:16,262 from these two particular trajectories, again this is just two examples. 133 00:10:16,262 --> 00:10:20,923 There are infinitely many such trajectories and infinitely many such 134 00:10:20,923 --> 00:10:23,798 term. But in any case, each of these terms is 135 00:10:23,798 --> 00:10:28,130 going to produce this sort of cosine of s1 minus s2 divided by h bar. 136 00:10:28,130 --> 00:10:31,580 So this is basically the quantum interference terms. 137 00:10:31,580 --> 00:10:36,873 And the terms which don't involve the cross product of different trajectories I 138 00:10:36,873 --> 00:10:40,562 classical terms. So in some sense, these are the terms. 139 00:10:40,562 --> 00:10:45,062 Which are responsible for the judic conductivity, the classical judic 140 00:10:45,062 --> 00:10:50,387 conductivity that we obtained in the previous discussion, okay and so now the 141 00:10:50,387 --> 00:10:55,262 question of what want what is, what is quantum mechanics doing, what are the 142 00:10:55,262 --> 00:10:59,474 quantum directions goes down to figuring out what this terms are. 143 00:10:59,474 --> 00:11:04,794 Now in the beginning of this video I mentioned something about temperature, 144 00:11:04,794 --> 00:11:09,202 said well at high temperature so this terms don't really matter at low 145 00:11:09,202 --> 00:11:12,850 temperatures. They start to measure and the reason this 146 00:11:12,850 --> 00:11:17,638 is the case because at high temperatures what the temperature really is, it 147 00:11:17,638 --> 00:11:22,882 involves lot of motion which occurs around these electrons which are sort of doing 148 00:11:22,882 --> 00:11:25,990 their thing moving around, diffusing around. 149 00:11:25,990 --> 00:11:31,006 But there is also let's say lattice vibrations, all kinds of things and this 150 00:11:31,006 --> 00:11:36,098 sort of shaking of the electron disturbs the phases, so in some sense this is what 151 00:11:36,098 --> 00:11:41,632 is called sometimes the quantum mechanical literature, this is called the phasing. 152 00:11:41,632 --> 00:11:46,724 So basically here in this expression whatever theory we're going to be doing 153 00:11:46,724 --> 00:11:51,664 we're going to determine these actions We're going to assume that phase one of 154 00:11:51,664 --> 00:11:56,604 mechanical phase is going to remain constant during the , or unperturbed more 155 00:11:56,604 --> 00:12:01,772 to say during the motion of election or along these trajectories, but it turns out 156 00:12:01,772 --> 00:12:06,408 that even if we assume that there is no disturbance of the phases and no new 157 00:12:06,408 --> 00:12:09,543 phasing. So classical physics in some sense 158 00:12:09,544 --> 00:12:15,431 protects itself in that this quantum interference theorems tends to cancel each 159 00:12:15,431 --> 00:12:18,806 other out and completely disappear from the picture. 160 00:12:18,806 --> 00:12:24,314 And we'll actually have to search really hard in order to find the [inaudible] 161 00:12:24,314 --> 00:12:29,696 where they start being essential. In order to see all this let me estimate 162 00:12:29,697 --> 00:12:33,030 the typical action that appears in this equation. 163 00:12:33,030 --> 00:12:36,440 So remember that. Well, at this stage, we're just talking 164 00:12:36,440 --> 00:12:39,639 about particles bouncing, bouncing around from these impurities. 165 00:12:39,639 --> 00:12:39,704 But otherwise, in between each scattering they are free particles. 166 00:12:39,704 --> 00:12:39,751 So their energy is essentially just mv squared over 2. 167 00:12:39,751 --> 00:12:39,817 The kinetic energy of a free particle. And therefore, in order to estimate the 168 00:12:39,817 --> 00:12:41,220 action. We can just say that well of the, by, by 169 00:12:41,220 --> 00:12:46,734 the at the projector. So now using this estimate we can also 170 00:12:46,734 --> 00:12:55,374 estimate the typical a quantum interference terms which are which I 171 00:12:55,374 --> 00:13:05,454 described by the sum of over all pairs of non-equivalent different trajectories L1 172 00:13:05,454 --> 00:13:15,534 and L2 and now instead of the action S we write exclusively at the typical momentum 173 00:13:15,534 --> 00:13:22,202 which by the way is called Feyman momentum divided by. 174 00:13:22,202 --> 00:13:24,602 H bar times the difference in length between the two trajectories under 175 00:13:24,602 --> 00:13:25,417 consideration. Or we can write the [inaudible] to be as 176 00:13:25,417 --> 00:13:25,840 the cosine sum over l one not equal to l two cosine of typical momentum u times 177 00:13:25,840 --> 00:13:29,960 delta l, divided by h bar. So it turns out that at these argumental, 178 00:13:29,960 --> 00:13:36,975 the cosine in the typical metal is actually very large and this is because 179 00:13:36,975 --> 00:13:44,335 the typical velocity of electron in a metal is less than the percent of the 180 00:13:44,335 --> 00:13:52,385 speed of light which I can't[INAUDIBLE] the speed of light which still is pretty 181 00:13:52,385 --> 00:14:00,435 fast and therefore the typical firmly to the typical wavelength turns out to be 182 00:14:00,435 --> 00:14:04,386 less than a. [unknown] minus nine meters, which is 183 00:14:04,387 --> 00:14:09,015 usually much smaller than the distance between these red dots here. 184 00:14:09,015 --> 00:14:14,251 So I'm just giving you facts, it's not by any means obvious, it doesn't follow from 185 00:14:14,251 --> 00:14:19,179 the previous discussion and I'm just giving you certain information and the 186 00:14:19,179 --> 00:14:24,184 bottom line here is that the typical wave length, in some sense you might think 187 00:14:24,184 --> 00:14:29,420 about action's really being a wave, and this wavelength is much, much smaller than 188 00:14:29,420 --> 00:14:34,665 the distance between these red dots. And so the interference terms are not very 189 00:14:34,665 --> 00:14:37,462 important. How it plays out here is that? 190 00:14:37,462 --> 00:14:42,742 So, this typical [inaudible] of delta l divided by essentially, this lambda is 191 00:14:42,742 --> 00:14:46,544 very large. And so, you have different trajectories 192 00:14:46,544 --> 00:14:50,600 contributing to this sum. And so you have cosines of different 193 00:14:50,600 --> 00:14:55,368 phases, completely random phases. And cosine as we know is the function 194 00:14:55,368 --> 00:15:00,452 which can be a positive or it can be negative depending so if this our cosine 195 00:15:00,452 --> 00:15:03,726 of pi, this is pi. So it can be positive or negative. 196 00:15:03,726 --> 00:15:07,542 So if we completely pick this phase is completely in random. 197 00:15:07,542 --> 00:15:10,638 So the average value of the cosine is going to be 0. 198 00:15:10,638 --> 00:15:15,606 So the terms with different signs of cosine basically cancel each other out and 199 00:15:15,606 --> 00:15:19,232 this is exactly. What happens with quantum interference 200 00:15:19,232 --> 00:15:22,448 terms? And so, seemingly classical physics 201 00:15:22,448 --> 00:15:27,648 survives, and we can just completely forget about this quantum interference 202 00:15:27,648 --> 00:15:31,052 phenom. So fortunately, it's not quite the case. 203 00:15:31,052 --> 00:15:36,819 It's almost the case, actually in [inaudible] disorder [inaudible] you see 204 00:15:36,819 --> 00:15:41,638 there is a leap year when an electron can actually travel in the clockwise 205 00:15:41,638 --> 00:15:45,359 direction, and this is going to be our path number one. 206 00:15:45,359 --> 00:15:50,765 Or it can decide to travel in the opposite direction, counterclockwise, in which case 207 00:15:50,765 --> 00:15:55,215 it's going to be a path number two. And the two paths, they're going to be 208 00:15:55,215 --> 00:15:59,205 different paths but they're going to have exactly the same way. 209 00:15:59,205 --> 00:16:04,437 And therefore the quantum interference terms responding to this pass are going to 210 00:16:04,437 --> 00:16:09,507 survive the this phase difference will be 0 and this terms are going to play 211 00:16:09,507 --> 00:16:14,499 [inaudible] significantly and so this is where we going to task and this second 212 00:16:14,499 --> 00:16:19,190 part of this of this video. Now going back to little bit to physics in 213 00:16:19,190 --> 00:16:23,155 order just to get give you an intuition of what, what, what it has to do with 214 00:16:23,155 --> 00:16:26,671 conductivity. So we can sort of can individually 215 00:16:26,671 --> 00:16:31,570 understand that the easier it is for an electron to move from an initial point to 216 00:16:31,570 --> 00:16:35,137 some just and final point battery, this material conducts. 217 00:16:35,138 --> 00:16:40,190 Okay the battery diffuses the battery it conducts and the classical level it is 218 00:16:40,190 --> 00:16:44,762 just [inaudible] by this [inaudible]. On the other hand the easier for the 219 00:16:44,762 --> 00:16:49,316 electron to return to the same point the, the more often it just sort of goes back 220 00:16:49,316 --> 00:16:52,040 and back. So the worst conductor it is. 221 00:16:52,040 --> 00:16:56,980 So in some sense the probability to have this self crossing trajectories, should in 222 00:16:56,980 --> 00:17:00,160 general, by general arguments reduce the conductivity. 223 00:17:00,160 --> 00:17:02,610 And this is exactly what happens, and we'll most cases. 224 00:17:02,610 --> 00:17:07,848 And therefore, it is called [inaudible]. Basically tries to bring electron back to 225 00:17:07,848 --> 00:17:08,943 where it came from.